Norm-Controlled Inversion in Smooth Banach Algebras, II

TL;DR

This paper demonstrates that in smooth Banach algebras, the property of smoothness is maintained under inversion, with the inverse's norm controlled by the original element's smoothness and spectral data, extending to ultra-smooth cases and matrix decay estimates.

Contribution

It establishes norm-controlled inversion for smooth elements in Banach algebras, including ultra-smooth cases and explicit decay estimates for matrices, advancing spectral analysis techniques.

Findings

Smoothness is preserved under inversion in Banach algebras.

Explicit norm control estimates are derived for matrices with polynomial decay.

The results extend classical approximation theory to operator algebras and matrix analysis.

Abstract

We show that smoothness implies norm-controlled inversion: the smoothness of an element $a$ in a Banach algebra with a one-parameter automorphism group is preserved under inversion, and the norm of the inverse $a^{-1}$ is controlled by the smoothness of $a$ and by spectral data. In our context smooth subalgebras are obtained with the classical constructions of approximation theory and resemble spaces of differentiable functions, Besov spaces or Bessel potential spaces. To treat ultra-smoothness, we resort to Dales-Davie algebras. Furthermore, based on Baskakov's work, we derive explicit norm control estimates for infinite matrices with polynomial off-diagonal decay. This is a quantitative version of Jaffard's theorem.